Optimal Order of One-Point and Multipoint Iteration
Journal of the ACM (JACM)
Three-step iterative methods with eighth-order convergence for solving nonlinear equations
Journal of Computational and Applied Mathematics
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A general class of $n$-point iterative methods for solving nonlinear equations is constructed by combining methods of Newton's type and an arbitrary two-point method of the fourth order of convergence. It is proved that these methods have the convergence order $2^n$, requiring only $n+1$ function evaluations per iteration. In this way it is demonstrated that the proposed class of methods supports the Kung-Traub hypothesis (1974) on the upper bound $2^n$ of the order of multipoint methods based on $n+1$ function evaluations. Consequently, this class possess as high as possible computational efficiency in the sense of the Kung-Traub hypothesis. Numerical examples are included to demonstrate exceptional convergence speed with only few function evaluations.